The internal energy of one gram of helium at 100 K and one atmospheric pressure is

Options

A. 100 J
B. 1200 J
C. 300 J
D. 500 J

Explanation

To find the internal energy of one gram of helium at 100 K and one atmospheric pressure, we can follow these steps:

### Step 1: Understand the formula for internal energy
The internal energy (U) of an ideal gas can be calculated using the formula:
\[ U = \frac{3}{2} nRT \]
where:
- \( n \) is the number of moles,
- \( R \) is the universal gas constant (approximately 8.31 J/mol·K),
- \( T \) is the temperature in Kelvin.

### Step 2: Calculate the number of moles of helium
The molar mass of helium (He) is approximately 4 g/mol. Since we have 1 gram of helium, we can calculate the number of moles (\( n \)) as follows:
\[ n = \frac{\text{mass of helium}}{\text{molar mass of helium}} = \frac{1 \text{ g}}{4 \text{ g/mol}} = \frac{1}{4} \text{ moles} \]

### Step 3: Substitute values into the internal energy formula
Now that we have the number of moles, we can substitute \( n \), \( R \), and \( T \) into the internal energy formula:
\[ U = \frac{3}{2} \left(\frac{1}{4}\right) R T \]
Substituting \( R = 8.31 \text{ J/mol·K} \) and \( T = 100 \text{ K} \):
\[ U = \frac{3}{2} \left(\frac{1}{4}\right) \times 8.31 \times 100 \]

### Step 4: Simplify the calculation
Calculating step-by-step:
1. Calculate \( \frac{3}{2} \times \frac{1}{4} = \frac{3}{8} \).
2. Now substitute this into the equation:
\[ U = \frac{3}{8} \times 8.31 \times 100 \]
3. First, calculate \( 8.31 \times 100 = 831 \).
4. Then, calculate \( \frac{3}{8} \times 831 = 311.625 \).

### Step 5: Round off the result
Since the question does not require a decimal point, we can round \( 311.625 \) to \( 312 \) joules.

### Final Answer
Thus, the internal energy of one gram of helium at 100 K and one atmospheric pressure is approximately:
\[ U \approx 312 \text{ joules} \]